Graphing Exponential Functions

Show your work for each problem.

1.
What is the y-intercept of the graph of $y = 5 \cdot 3^x$ ?
1. a) $(0, 15)$
2. b) $(5, 0)$
3. c) $(0, 5)$
4. d) $(0, 3)$
Answer: $(0, 5)$
$y = 5 \cdot 3^0 = 5 \cdot 1 = 5$ , so the graph crosses the y-axis at $(0, 5)$ . The base 3 only decides how fast the graph rises.
2.
Which graph falls from left to right?
1. a) $y = 0.5 \cdot 3^x$
2. b) $y = 2 \cdot (1.1)^x$
3. c) $y = 4 \cdot (0.3)^x$
4. d) $y = 4 \cdot 2^x$
Answer: $y = 4 \cdot (0.3)^x$
All four have $a > 0$ , so the base decides. Only $0.3$ is between 0 and 1, so $y = 4 \cdot (0.3)^x$ falls. The factor $0.5$ in front of $3^x$ only compresses the graph; it still rises because $3 > 1$ .
3.
What is the horizontal asymptote of $y = 2^x - 7$ ? Enter the number $k$ in $y = k$ .
Answer: -7
Subtracting 7 moves the whole graph 7 units down, including its asymptote: $y = 0$ becomes $y = -7$ .
4.
For $f(x) = 3 \cdot 2^x$ , find $f(-1)$ . Give your answer as a decimal or a fraction.
Answer: 1.5
$f(-1) = 3 \cdot 2^{-1} = 3 \cdot \frac{1}{2} = 1.5$ . The point $(-1, 1.5)$ lies on the flat left part of the graph.
5.
Make a table of values for $y = 4 \cdot 2^x$ . What is $y$ at $x = 3$ ?
Answer: 32
1. What is $y$ at $x = -2$ ? 1
2. What is $y$ at $x = 0$ ? 4
3. What is $y$ at $x = 2$ ? 16
4. What is $y$ at $x = 3$ ? 32
6.
Make a table of values for $y = 6 \cdot (0.5)^x$ . What is $y$ at $x = 3$ ? Give a decimal.
Answer: 0.75
1. What is $y$ at $x = -2$ ? 24
2. What is $y$ at $x = -1$ ? 12
3. What is $y$ at $x = 1$ ? 3
4. What is $y$ at $x = 3$ ? 0.75
7.
The graph of $y = 2^x$ is shifted 3 units left and 1 unit down. Which equation describes the new graph?
1. a) $y = 2^{x-1} + 3$
2. b) $y = 2^{x+3} + 1$
3. c) $y = 2^{x-3} - 1$
4. d) $y = 2^{x+3} - 1$
Answer: $y = 2^{x+3} - 1$
Left 3 means $x - h$ with $h = -3$ , that is $x + 3$ ; down 1 means $k = -1$ . So $y = 2^{x+3} - 1$ , with asymptote $y = -1$ .
8.
What is the range of $y = -4 \cdot 2^x + 1$ ?
1. a) $y > -4$
2. b) $y < 1$
3. c)all real numbers
4. d) $y > 1$
Answer: $y < 1$
The asymptote is $y = 1$ ( $k = 1$ ). Since $a = -4 < 0$ , the term $-4 \cdot 2^x$ is always negative, so $y$ is always less than 1: the range is $y < 1$ .
9.
The graph of $y = 3^{x+2} - 5$ crosses the y-axis at $(0, ?)$ . Enter the y-coordinate.
Answer: 4
$y = 3^{0+2} - 5 = 9 - 5 = 4$ . The y-intercept is $(0, 4)$ , not $(0, 1)$ and not $(0, -4)$ , because the graph was also shifted 2 units left.
10.
The graph of $y = a \cdot b^x$ passes through $(0, 5)$ and $(2, 45)$ , with $b > 0$ . Find $b$ .
Answer: 3
$a = 5$ from the y-intercept. Then $5 \cdot b^2 = 45$ , so $b^2 = 9$ and $b = 3$ (the base is positive). The equation is $y = 5 \cdot 3^x$ .
11.
The graph of $y = a \cdot b^x$ passes through $(0, 3)$ and $(1, 6)$ . Find the equation and the value of $y$ at $x = 4$ .
Answer: 48
1. What is $a$ ? 3
2. What is $b$ ? 2
3. What is $y$ at $x = 4$ ? 48
12.
For $y = 2^{x+1} - 4$ , find the asymptote, the y-intercept and the x-intercept. Where does the graph cross the x-axis? Enter the x-coordinate of that point.
Answer: 1
1. The asymptote is $y = k$ . What is $k$ ? -4
2. What is $y$ at $x = 0$ ? -2
3. Solve $2^{x+1} = 4$ . What is $x$ ? 1
13.
Make a table for $y = -3 \cdot 2^x + 2$ and find its asymptote. Give decimals where needed.
Answer: 2
1. What is $y$ at $x = 0$ ? -1
2. What is $y$ at $x = -1$ ? 0.5
3. What is $y$ at $x = 2$ ? -10
4. The asymptote is $y = k$ . What is $k$ ? 2
14.
Which description matches the graph of $y = -2 \cdot 3^{x-1} + 5$ ?
1. a)Falling, asymptote $y = -5$ , range $y < -5$
2. b)Rising, asymptote $y = 1$ , range $y < 1$
3. c)Rising, asymptote $y = 5$ , range $y > 5$
4. d)Falling, asymptote $y = 5$ , range $y < 5$
Answer: Falling, asymptote $y = 5$ , range $y < 5$
$k = 5$ gives the asymptote $y = 5$ . Since $a = -2 < 0$ , the graph is below the asymptote, so the range is $y < 5$ . With $b = 3 > 1$ , the term $3^{x-1}$ grows, and multiplying by $-2$ makes $y$ decrease from left to right.
15.
The graph of $y = a \cdot b^x$ passes through $(1, 6)$ and $(3, 54)$ , with $b > 0$ . Find $a$ .
Answer: 2
$b^2 = 54 \div 6 = 9$ , so $b = 3$ . Then $a \cdot 3 = 6$ gives $a = 2$ . Check: $2 \cdot 3^3 = 54$ . The equation is $y = 2 \cdot 3^x$ .
16.
The graph of $y = a \cdot b^x$ passes through $(2, 12)$ and $(4, 48)$ , with $b > 0$ . Find the equation and the value of $y$ at $x = 5$ .
Answer: 96
1. What is $b^2 = 48 \div 12$ ? 4
2. What is $b$ ? 2
3. What is $a$ ? 3
4. What is $y$ at $x = 5$ ? 96
17.
A graph of the form $y = a \cdot b^x + 2$ has the asymptote $y = 2$ and passes through $(0, 5)$ and $(1, 11)$ . Find $a$ and $b$ , then the value of $y$ at $x = 2$ .
Answer: 29
1. Use $(0, 5)$ : $a \cdot 1 + 2 = 5$ . What is $a$ ? 3
2. Use $(1, 11)$ : $3 \cdot b + 2 = 11$ . What is $b$ ? 3
3. What is $y$ at $x = 2$ ? 29
18.
Find where the graph of $y = 3 \cdot 2^x - 24$ crosses both axes. What is the x-coordinate of the x-intercept?
Answer: 3
1. What is $y$ at $x = 0$ ? -21
2. Set $y = 0$ : $3 \cdot 2^x = 24$ . What is $2^x$ ? 8
3. What is $x$ ? 3